chore(data/nat/modeq): split out lemmas about lists (#18004)

By splitting out some lemmas relating list-rotation and modular arithmetic, the files `data.nat.modeq` and `data.int.modeq` become portable now.

Author: hrmacbeth

archive/miu_language/decision_nec.lean

@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Gihan Marasingha
 -/
 import .basic
+import data.list.count
 import data.nat.modeq
 import tactic.ring
 

src/data/list/modeq.lean

@@ -0,0 +1,68 @@
+/-
+Copyright (c) 2018 Chris Hughes. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Chris Hughes
+-/
+import data.nat.modeq
+import data.list.rotate
+
+/-! # List rotation and modular arithmetic -/
+
+namespace list
+variable {α : Type*}
+
+lemma nth_rotate : ∀ {l : list α} {n m : ℕ} (hml : m < l.length),
+  (l.rotate n).nth m = l.nth ((m + n) % l.length)
+| []     n     m hml := (nat.not_lt_zero _ hml).elim
+| l      0     m hml := by simp [nat.mod_eq_of_lt hml]
+| (a::l) (n+1) m hml :=
+have h₃ : m < list.length (l ++ [a]), by simpa using hml,
+(lt_or_eq_of_le (nat.le_of_lt_succ $ nat.mod_lt (m + n)
+  (lt_of_le_of_lt (nat.zero_le _) hml))).elim
+(λ hml',
+  have h₁ : (m + (n + 1)) % ((a :: l : list α).length) =
+      (m + n) % ((a :: l : list α).length) + 1,
+    from calc (m + (n + 1)) % (l.length + 1) =
+      ((m + n) % (l.length + 1) + 1) % (l.length + 1) :
+        add_assoc m n 1 ▸ nat.modeq.add_right 1 (nat.mod_mod _ _).symm
+    ... = (m + n) % (l.length + 1) + 1 : nat.mod_eq_of_lt (nat.succ_lt_succ hml'),
+  have h₂ : (m + n) % (l ++ [a]).length < l.length, by simpa [nat.add_one] using hml',
+  by rw [list.rotate_cons_succ, nth_rotate h₃, list.nth_append h₂, h₁, list.nth]; simp)
+(λ hml',
+  have h₁ : (m + (n + 1)) % (l.length + 1) = 0,
+    from calc (m + (n + 1)) % (l.length + 1) = (l.length + 1) % (l.length + 1) :
+      add_assoc m n 1 ▸ nat.modeq.add_right 1
+        (hml'.trans (nat.mod_eq_of_lt (nat.lt_succ_self _)).symm)
+    ... = 0 : by simp,
+  by rw [list.length, list.rotate_cons_succ, nth_rotate h₃, list.length_append,
+    list.length_cons, list.length, zero_add, hml', h₁, list.nth_concat_length]; refl)
+
+lemma rotate_eq_self_iff_eq_repeat [hα : nonempty α] : ∀ {l : list α},
+  (∀ n, l.rotate n = l) ↔ ∃ a, l = list.repeat a l.length
+| []     := ⟨λ h, nonempty.elim hα (λ a, ⟨a, by simp⟩), by simp⟩
+| (a::l) :=
+⟨λ h, ⟨a, list.ext_le (by simp) $ λ n hn h₁,
+  begin
+    rw [← option.some_inj, ← list.nth_le_nth],
+    conv {to_lhs, rw ← h ((list.length (a :: l)) - n)},
+    rw [nth_rotate hn, add_tsub_cancel_of_le (le_of_lt hn),
+      nat.mod_self, nth_le_repeat], refl
+  end⟩,
+  λ ⟨a, ha⟩ n, ha.symm ▸ list.ext_le (by simp)
+    (λ m hm h,
+      have hm' : (m + n) % (list.repeat a (list.length (a :: l))).length < list.length (a :: l),
+        by rw list.length_repeat; exact nat.mod_lt _ (nat.succ_pos _),
+      by rw [nth_le_repeat, ← option.some_inj, ← list.nth_le_nth, nth_rotate h, list.nth_le_nth,
+        nth_le_repeat]; simp * at *)⟩
+
+lemma rotate_repeat (a : α) (n : ℕ) (k : ℕ) :
+  (list.repeat a n).rotate k = list.repeat a n :=
+let h : nonempty α := ⟨a⟩ in by exactI rotate_eq_self_iff_eq_repeat.mpr ⟨a, by rw length_repeat⟩ k
+
+lemma rotate_one_eq_self_iff_eq_repeat [nonempty α] {l : list α} :
+  l.rotate 1 = l ↔ ∃ a : α, l = list.repeat a l.length :=
+⟨λ h, rotate_eq_self_iff_eq_repeat.mp (λ n, nat.rec l.rotate_zero
+  (λ n hn, by rwa [nat.succ_eq_add_one, ←l.rotate_rotate, hn]) n),
+    λ h, rotate_eq_self_iff_eq_repeat.mpr h 1⟩
+
+end list

src/data/nat/modeq.lean

@@ -4,7 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Mario Carneiro
 -/
 import data.int.gcd
-import data.list.rotate
 import tactic.abel
 
 /-!
@@ -425,62 +424,3 @@ have help : ∀ (m : ℕ), m < 4 → m % 2 = 1 → m = 1 ∨ m = 3 := dec_trivia
  λ h, or.dcases_on h odd_of_mod_four_eq_one odd_of_mod_four_eq_three⟩
 
 end nat
-
-namespace list
-variable {α : Type*}
-
-lemma nth_rotate : ∀ {l : list α} {n m : ℕ} (hml : m < l.length),
-  (l.rotate n).nth m = l.nth ((m + n) % l.length)
-| []     n     m hml := (nat.not_lt_zero _ hml).elim
-| l      0     m hml := by simp [nat.mod_eq_of_lt hml]
-| (a::l) (n+1) m hml :=
-have h₃ : m < list.length (l ++ [a]), by simpa using hml,
-(lt_or_eq_of_le (nat.le_of_lt_succ $ nat.mod_lt (m + n)
-  (lt_of_le_of_lt (nat.zero_le _) hml))).elim
-(λ hml',
-  have h₁ : (m + (n + 1)) % ((a :: l : list α).length) =
-      (m + n) % ((a :: l : list α).length) + 1,
-    from calc (m + (n + 1)) % (l.length + 1) =
-      ((m + n) % (l.length + 1) + 1) % (l.length + 1) :
-        add_assoc m n 1 ▸ nat.modeq.add_right 1 (nat.mod_mod _ _).symm
-    ... = (m + n) % (l.length + 1) + 1 : nat.mod_eq_of_lt (nat.succ_lt_succ hml'),
-  have h₂ : (m + n) % (l ++ [a]).length < l.length, by simpa [nat.add_one] using hml',
-  by rw [list.rotate_cons_succ, nth_rotate h₃, list.nth_append h₂, h₁, list.nth]; simp)
-(λ hml',
-  have h₁ : (m + (n + 1)) % (l.length + 1) = 0,
-    from calc (m + (n + 1)) % (l.length + 1) = (l.length + 1) % (l.length + 1) :
-      add_assoc m n 1 ▸ nat.modeq.add_right 1
-        (hml'.trans (nat.mod_eq_of_lt (nat.lt_succ_self _)).symm)
-    ... = 0 : by simp,
-  by rw [list.length, list.rotate_cons_succ, nth_rotate h₃, list.length_append,
-    list.length_cons, list.length, zero_add, hml', h₁, list.nth_concat_length]; refl)
-
-lemma rotate_eq_self_iff_eq_repeat [hα : nonempty α] : ∀ {l : list α},
-  (∀ n, l.rotate n = l) ↔ ∃ a, l = list.repeat a l.length
-| []     := ⟨λ h, nonempty.elim hα (λ a, ⟨a, by simp⟩), by simp⟩
-| (a::l) :=
-⟨λ h, ⟨a, list.ext_le (by simp) $ λ n hn h₁,
-  begin
-    rw [← option.some_inj, ← list.nth_le_nth],
-    conv {to_lhs, rw ← h ((list.length (a :: l)) - n)},
-    rw [nth_rotate hn, add_tsub_cancel_of_le (le_of_lt hn),
-      nat.mod_self, nth_le_repeat], refl
-  end⟩,
-  λ ⟨a, ha⟩ n, ha.symm ▸ list.ext_le (by simp)
-    (λ m hm h,
-      have hm' : (m + n) % (list.repeat a (list.length (a :: l))).length < list.length (a :: l),
-        by rw list.length_repeat; exact nat.mod_lt _ (nat.succ_pos _),
-      by rw [nth_le_repeat, ← option.some_inj, ← list.nth_le_nth, nth_rotate h, list.nth_le_nth,
-        nth_le_repeat]; simp * at *)⟩
-
-lemma rotate_repeat (a : α) (n : ℕ) (k : ℕ) :
-  (list.repeat a n).rotate k = list.repeat a n :=
-let h : nonempty α := ⟨a⟩ in by exactI rotate_eq_self_iff_eq_repeat.mpr ⟨a, by rw length_repeat⟩ k
-
-lemma rotate_one_eq_self_iff_eq_repeat [nonempty α] {l : list α} :
-  l.rotate 1 = l ↔ ∃ a : α, l = list.repeat a l.length :=
-⟨λ h, rotate_eq_self_iff_eq_repeat.mp (λ n, nat.rec l.rotate_zero
-  (λ n hn, by rwa [nat.succ_eq_add_one, ←l.rotate_rotate, hn]) n),
-    λ h, rotate_eq_self_iff_eq_repeat.mpr h 1⟩
-
-end list

src/group_theory/perm/cycle/type.lean

@@ -6,6 +6,7 @@ Authors: Thomas Browning
 
 import algebra.gcd_monoid.multiset
 import combinatorics.partition
+import data.list.modeq
 import group_theory.perm.cycle.basic
 import ring_theory.int.basic
 import tactic.linarith